A Bose condensate subject to a periodic modulation of the two-body interactions was recently observed to emit matter-wave jets resembling "fireworks" [Nature 551, 356(2017)]. In this paper, combining experiment with numerical simulation, we demonstrate that these "Bose fireworks" represent a late stage in a complex time evolution of the driven condensate. We identify a "density wave" stage which precedes jet emission and results from interference of matterwaves. The density waves self-organize and self-amplify without the breaking of long range translational symmetry. Importantly, this density wave structure deterministically establishes the template for the subsequent patterns of the emitted jets. Our simulations, in good agreement with experiment, also address the apparent asymmetry in the jet pattern and show it is fully consistent with momentum conservation.Time-periodic driving, which allows coherent manipulation of many-body systems, is becoming an exciting tool in the ultracold atomic gases. This provides access to new quantum physics, for example, topological states, synthetic gauge fields and Mott transitions [1][2][3][4][5]. Of particular interest is the rather unique capability these atomic systems afford into understanding non-equilibrium many-body dynamics [6]. Also unique to the ultracold gases is the ability, through the Feshbach resonance, to periodically modulate atomic interactions [7]. Recently, this was implemented by the Chicago group [8,9] and the Rice group [10-12] on Bose-Einstein condensates. In the Chicago experiment, a collective emission of matter-wave jets resembling fireworks occurs above a threshold modulation amplitude.The jets were associated with a form of runaway stimulated inelastic scattering occurring in the driven condensate [8].In this paper we use the time-dependent Gross-Pitaevskii (GP) equation to study the evolution of the modulated BEC and the emission of these jets [8]. An unbiased or random noise term is introduced initially to model the fluctuations that seed the jet emission. We show that the simulations capture well the "fireworks" dynamics seen in experiments. Moreover, in combination with a new set of experiments, we identify a previously unobserved stage of the evolution that precedes and underlies the jet-emission. Immediately after modulation, we observe that density waves emerge and grow rapidly in the condensate with quantized wavenumbers determined by the modulation frequency [13]. The density waves arise from the interference between excited matterwaves and the condensate. The pattern is reminiscent of Faraday waves in nonlinear fluids [14,15] and related to that predicted for driven atomic gases [16][17][18][19] as well as observed in the one-dimensional condensate [20].The amplification of these density waves can be considered the matterwave analog of superradiant 30 µm 0 5 9 14 18 t = 28 ms Time t Experiment Simulation Density n (µm -2 ) 0 30 15 Density n/ n 0 0 0.6 0.3FIG. 1. The real space density distribution n(r) (denoted as n) as a comparison betw...